Final answer:
To solve the equation 2⁷ˣ-7=4 without using logarithms, rewrite the equation using the properties of exponents and evaluate the exponent before solving for x, which gives us the solution x = 0.085 (rounded to three decimal places).
Step-by-step explanation:
To solve the exponential equation 2⁷ˣ-7=4 without using logarithms, we can rewrite the equation using the properties of exponents. Note that 2⁷ˣ can also be written as (2⁷)ˣ. So the equation becomes (2⁷)ˣ-7 = 4. Now we can simplify the equation by evaluating 2⁷ = 128, so the equation becomes 128ˣ-7 = 4. Add 7 to both sides of the equation to isolate the exponent, giving us 128ˣ = 11. Divide both sides of the equation by 128 to solve for x, which gives us x = 0.085. Therefore, the solution to the equation is x = 0.085 (rounded to three decimal places).